5-demicube

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5-demicube
Demipenteract
(5-demicube) Petrie polygon projection
Type Uniform 5-polytope
Family (Dn) 5-demicube
Families (En) k21 polytope
1k2 polytope
Coxeter symbol 121
Schläfli symbol {31,2,1}
h{4,3,3,3}
s{2,2,2,2}
Coxeter-Dynkin diagram                         4-faces 26 10 {31,1,1} 16 {3,3,3} Cells 120 40 {31,0,1} 80 {3,3} Faces 160 {3} Edges 80
Vertices 16
Vertex figure rectified 5-cell
Petrie polygon Octagon
Symmetry group D5, [34,1,1] = [1+,4,33]
+
Properties convex

In five dimensional geometry, a demipenteract or 5-demicube is a semiregular 5-polytope, constructed from a 5-hypercube (penteract) with alternated vertices deleted.

It was discovered by Thorold Gosset. Since it was the only semiregular 5-polytope (made of more than one type of regular hypercell), he called it a 5-ic semi-regular.

Coxeter named this polytope as 121 from its Coxeter-Dynkin diagram, which has branches of length 2, 1 and 1 with a ringed node on one of the short branches. It exists in the k21 polytope family as 121 with the Gosset polytopes: 221, 321, and 421.

Cartesian coordinates

Cartesian coordinates for the vertices of a demipenteract centered at the origin and edge length 2√2 are alternate halves of the penteract:

(±1,±1,±1,±1,±1)

with an odd number of plus signs.

Projected images Perspective projection.

Images

orthographic projections
Coxeter plane B5
Graph Dihedral symmetry [10/2]
Coxeter plane D5 D4
Graph  Dihedral symmetry  
Coxeter plane D3 A3
Graph  Dihedral symmetry  

Related polytopes

It is a part of a dimensional family of uniform polytopes called demihypercubes for being alternation of the hypercube family.

There are 23 uniform polytera (uniform 5-polytope) that can be constructed from the D5 symmetry of the demipenteract, 8 of which are unique to this family, and 15 are shared within the penteractic family.

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